HW7_Sol - Math 104 Fall 2009 Homework 7 solution set We use...

Math 104 Fall 2009 Homework 7: solution set We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, A is a matrix and a ij its ( i,j )th entry. Likewise x is a vector and x j its j th component. The linear span generated by a group of vecotors v 1 , v 2 ,..., v n is denoted by span( v 1 , v 2 ,..., v n ). Problem 1 Since the characteristic polynomial is P ( λ ) = ( λ-λ 1 ) d 1 ··· ( λ-λ k ) d k , we know all the eigenvalues are λ 1 , ··· ,λ k with corresponding multiplicity d 1 , ··· ,d k . Therefore, since the trace is the sum of the eigenvalues, trace( A ) = ∑ k i =1 d i λ i , and since the determinant is the product of the eigenvalues, det( A ) = Q k i =1 λ d i i . Problem 2 (a) Set B = i A . Since A * =-A , B * =-i A * = i A = B , which implies that B is Hermitian. By the spectral theorem, there is a unitary matrix U and a real diagonal matrix Λ such that B = U Λ

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